The Dold-Kan Correspondence and Coalgebra Structures

TL;DR

This paper establishes a Quillen adjunction and equivalence between categories of non-cocommutative coalgebras in simplicial and differential graded contexts using the Dold-Kan correspondence.

Contribution

It constructs a Quillen adjunction and proves a Quillen equivalence for connected coalgebras, extending the Dold-Kan correspondence to coalgebra structures.

Findings

Established a Quillen adjunction between coalgebra categories

Proved a Quillen equivalence for connected coalgebras

Extended Dold-Kan correspondence to non-cocommutative coalgebras

Abstract

By using the Dold-Kan correspondence we construct a Quillen adjunction between the model categories of non-cocommutative coassociative simplicial and differential graded coalgebras over a field. We restrict to categories of connected coalgebras and prove a Quillen equivalence between them.